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In this paper we will deduce several properties of the Green's functionsrelated to the Hill's equation coupled to various boundary value conditions. Inparticular, the idea is to study the Green's functions of the second orderdifferential operator coupled to Neumann, Dirichlet, Periodic and Mixedboundary conditions, by expressing the Green's function of a given problem as alinear combination of the Green's function of the other ones. This will allowus to compare different Green's functions when they have constant sign.Finally, such properties of the Green's function of the linear problem will befundamental to deduce the existence of solutions to the nonlinear problem. Theresults are derived from the fixed point theory applied to related operatorsdefined on suitable cones in Banach spaces.

Relationship of the Green's functions related to the Hill's equation coupled to different boundary value conditions